RH-W-13: Cross-Regularity Continuation and Canonical Parameters

The explorer once reported a suspected Weil negative direction of -3.32×10⁻⁷ — which, if true, would be a major counterexample. This package neither publicized nor immediately deleted it, but instead upgraded it to a red alert candidate for layer-by-layer investigation according to protocol. The root cause was found to be that the parameters of M were cache-quantized while G was not synchronously quantized, causing an "object identity error" in the generalized eigenvalue problem. After consistent quantization, it flipped positive to +8.76×10⁻⁸, which was confirmed by 80-digit independent integration. Finally, using a jump-resolved tail bound, the exact 10-dimensional mixed generalized spectral bottom was squeezed into (10⁻⁸, 5×10⁻⁸).

RH_CLAIM = False — The claim status output by the verifier within the package itself, reproduced as is. The near-zero positive spectral band is neither an RH proof nor a counterexample; false negatives have been explicitly marked as NUMERICAL_FALSE_NEGATIVE_REJECTED, not quietly removed.

Connections

Regarding its relationship with other packages, I try to use the words from its own documents, not my interpretation.

Belongs toResearch Starting Point v1.6 · Cross-Regularity Near-Zero Spectral Band

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"If this is a strictly negative direction of the same real Weil dictionary, it would be of extreme significance. Therefore, the correct engineering behavior is neither to publicize it nor to delete it immediately, but to escalate it to a red-alert candidate and investigate it layer by layer... In computational mathematics, the version consistency of shared metrics is part of the theorem's input, not a software detail." — Excerpt from Document 02 in this package, "Consistent Quantization Counterexample and Red Alert Audit."

GAP Ledger

From RH-W-13_subgaps_v0.1.csv within the package; the status and descriptions are recorded by the package itself, not my retrospective judgment.

Reproduce

Cross-regularity certificate construction → purely rational exact verifier → 80-digit mpmath independent cross-check:

python build_cross_regularity_certificate.py
python verify_cross_regularity.py
python crosscheck_w13_mpmath.py

01 · Cross-Regularity Continuation and Canonical Parameters

Main document — why α is merely a gauge/preconditioning parameter rather than a true continuation direction, the true continuation parameter σ, the consistent parameter principle, the selected rational chamber, the exact spectral bracket (10⁻⁸, 5×10⁻⁸), and the dual superposition of channel cancellation and arithmetic cancellation.

02 · Uniform Quantization Counterexample and Red Alert Audit

Why this failure must be preserved, the erroneous structure of M/G quantization inconsistency, the three-way result comparison table (pre-quantization / post-consistent quantization / 80-bit crosscheck), new mandatory rules, and the distinction between "object identity error" in generalized eigenvalue problems and general matrix errors.

03 · Jump-Resolved Tail Bounds and Near-Zero Certificates

Why the old tail bound could not resolve the 10⁻⁸ level spectral bottom, exact jump representation, extra decay of the exponential jump tail bound, cutoff and interval width of this round, and the final upper and lower bound certificate.

Verification Output

From ALL_VERIFY.txt in the package (merged output of exact verifier + mpmath cross-check, including the interval width for each block-lag), reproduced line by line.

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Files

The raw JSON of the quantization mismatch audit (quantization_mismatch_audit.json) and the continuation search paths are not listed individually; they are all in the full zip file.

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